* Kiev vows to press on with its anti-terrorist campaign (Adds detail)

By Sabina Zawadzki and Gabriela Baczynska

DONETSK, Ukraine, May 27 (Reuters) - Ukrainian forces fought with separatists in the city of Donetsk for a second day on Tuesday after inflicting heavy losses on the rebels and the government vowed to press on with a military offensive "until not a single terrorist" was left.

Pro-Russian rebels said more than 50 of their fighters had been killed. The mayor of Donetsk, an industrial hub of one million in eastern Ukraine, said the death toll in the clashes which erupted on Monday stood at 48, including two civilians.

A Reuters correspondent counted 20 bodies in combat fatigues in one room of a hospital morgue, some of them missing limbs. Rebels said they had all been killed in a truck that came under fierce attack by the army.

"From our side, there are more than 50 (dead)," the prime minister of the rebels' self-styled Donetsk People's Republic, Alexander Borodai, told Reuters at the hospital, adding that Ukrainian troops were now in control of the airport.

Ukraine launched its fresh offensive against the rebels, who have occupied strategic points in Donetsk and other towns in the east, shortly after Sunday's presidential election. The election victor, Petro Poroshenko, rejected any talks with "terrorists".

Kiev accused Moscow on Tuesday of sending "terrorists" across the border after border guards clashed with armed men overnight in eastern Ukraine. Russia repeated its call on Ukraine to halt its military operations.

Ukraine used air strikes and a paratroop assault on Monday to clear rebels from Donetsk's international terminal and had pushed the separatists out of the complex by the end of the day.

But shooting continued through the night and on Tuesday the road to the airport bore signs of fighting overnight and heavy machinegun fire could be heard in the distance in mid-morning.

"The airport is completely under control," Interior Minister Arsen Avakov told journalists in the capital Kiev. "The adversary suffered heavy losses. We have no losses," he added.

"We'll continue the anti-terrorist operation until not a single terrorist remains on the territory of Ukraine," First Deputy Prime Minister Vitaly Yarema said on the margins of a government meeting.

The Organization for Security and Cooperation in Europe (OSCE) said it had lost contact with a monitoring team in the Donetsk region. The team, comprising an Estonian, a Swiss, a Turk and a Dane, met a road checkpoint on Monday night at around 6 p.m. and had not re-established contact.

EXECUTIONS

Donetsk mayor Alexander Lukyanchenko said 48 people had been killed and 43 injured in the past day.

Donetsk police said the rebels had killed two policemen on Tuesday in the nearby town of Gorlovka. The Ostrov news website said the men had been executed with sacks over their heads for "breaking their oath to the Donetsk People's Republic".

In Donetsk city, an industrial hub of one million, posters went up in the streets telling people how to act if shooting broke out and where bomb shelters were located. Schools were closed and parents were told to keep their children indoors.

It was too early to say if the apparent military successes of government forces could break the rebellion in Donetsk, the first of a rash of such revolts across eastern Ukraine after months of turmoil in the capital Kiev that led to the ousting of Ukraine's Moscow-backed president, Viktor Yanukovich.

Despite their losses, separatists on Tuesday chopped down trees and used vehicles and sandbags to build a new barricade across the road to the airport.

Ukrainians hope that the resounding victory of Poroshenko, a 48-year-old billionaire businessman, in Sunday's election will help to draw a line under six months of political upheaval in the former Soviet republic. He said a robust military campaign in the east could quash the separatists in "a matter of hours".

On the road to Donetsk airport, Reuters journalists saw blood-spattered and bullet-riddled hulks of trucks of the kind the separatist fighters have been using.

A Kamaz truck lay wrecked on the airport road, covered in blood and completely riddled with bullet holes. Blood had spattered across the entire roadway and even reached a billboard seven metres above. Outside the city centre, another Kamaz was flipped over, its deck covered in blood.

It was in one such vehicle that the 20 dead had been when it came under attack. "They shot at them from the ground and from the air," said a separatist who gave his name only as Alexander.

In Donetsk on Tuesday a hockey stadium was set ablaze, though firefighters managed to bring it under control.

The mayor reinforced warnings for people to avoid the airport area and said nine schools and two hospitals had been closed, with patients evacuated elsewhere. He reassured people that food and other supplies were still reaching the city.

Gleb, an engineer in his early 40s, said people were hiding in their homes, horrified at the chaos in their city.

"These separatists have made a big mess of the city. They probably hoped (Russian President Vladimir) Putin would come in with help but now they seem to be left on their own," he said.

"Now things have gone so far, it will be very difficult for things to get back to normal afterwards." (Writing by Richard Balmforth; Editing by Gareth Jones)